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In recent years, the inconsistency in Bayesian deep learning has garnered increasing attention. Tempered or generalized posterior distributions often offer a direct and effective solution to this issue. However, understanding the underlying causes and evaluating the effectiveness of generalized posteriors remain active areas of research. In this study, we introduce a unified theoretical framework to attribute Bayesian inconsistency to model misspecification and inadequate priors. We interpret the generalization of the posterior with a temperature factor as a correction for misspecified models through adjustments to the joint probability model, and the recalibration of priors by redistributing probability mass on models within the hypothesis space using data samples. Additionally, we highlight a distinctive feature of Laplace approximation, which ensures that the generalized normalizing constant can be treated as invariant, unlike the typical scenario in general Bayesian learning where this constant varies with model parameters post-generalization. Building on this insight, we propose the generalized Laplace approximation, which involves a simple adjustment to the computation of the Hessian matrix of the regularized loss function. This method offers a flexible and scalable framework for obtaining high-quality posterior distributions. We assess the performance and properties of the generalized Laplace approximation on state-of-the-art neural networks and real-world datasets., Comment: The paper has some theoretical flaws and will be updated after correction

In this study, we propose a decision-making strategy for pinning-based distributed multi-agent (PDMA) automatic generation control (AGC) in islanded microgrids against stochastic communication disruptions. The target microgrid is construed as a cyber-physical system, wherein the physical microgrid is modeled as an inverter-interfaced autonomous grid with detailed system dynamic formulation, and the communication network topology is regarded as a cyber-system independent of its physical connection. The primal goal of the proposed method is to decide the minimum number of generators to be pinned and their identities amongst all distributed generators (DGs). The pinningdecisions are made based on complex network theories using the genetic algorithm (GA), for the purpose of synchronizing and regulating the frequencies and voltages of all generator busbars in a PDMA control structure, i.e., without resorting to a central AGC agent. Thereafter, the mapping of cyber-system topology and the pinning decision is constructed using deeplearning (DL) technique, so that the pinning-decision can be made nearly instantly upon detecting a new cyber-system topology after stochastic communication disruptions. The proposed decision-making approach is verified using a 10-generator, 38-bus microgrid through time-domain simulation for transient stability analysis., Comment: 8 pages, 7 figures, 2 tables